Question 89
For an acute triangle $ABC$: its circumcircle $\omega$ with center $O$, the circumcircle $\omega_1$ of the triangle $AOC$ and the diameter $OQ$ of $\omega_1$ were drawn. The points $M$ and $N$ were taken on the lines $AQ$ and $AC$ respectively in such a way that the quadrilateral $AMBN$ is a parallelogram. Prove that the intersection point of the lines $MN$ and $BQ$ belongs to the circle $\omega_1$.